Proof. Idempotents are preserved by automorphisms, and principal idempotents of L can be characterised as those $e \in \text {Idem}(L)$ for which $ef = e$ or $ef = 0$ for any $f \in \text {Idem}(L)$ .

Proof. Fix some $e_0 \in \{e_i\}_i$ with $\dim _{L_0} e_0 \cdot M < \infty $ , and for each i fix some $g_i \in G$ with $g_i(e_0) = e_i$ . This allows us to define field isomorphisms

where $\pi _i \colon L \rightarrow L_i$ is the projection and $\iota _0 \colon L_0 \rightarrow L$ is defined by setting $\iota _0(\lambda )$ to be the unique element of L with $\pi _0(\iota _0(\lambda )) = \lambda $ and $\pi _j (\iota _0(\lambda )) = 0$ when $j \neq 0$ .

The map $g_i \colon M \rightarrow M$ restricts to a bijection $e_0 \cdot M \xrightarrow {\sim } e_i \cdot M$ , which is $L_0$ -linear with respect to the natural action of $L_0$ on $e_0 \cdot M$ and the action of $L_0$ on $e_i \cdot M$ through $\gamma _i \colon L_0 \xrightarrow {\sim } L_i$ . Then if $\{f_m^0\}_m$ is a basis for $e_0 \cdot M$ over $L_0$ , the set $\{f_m\}_m$ defined by $f_m = (g_i(f_m^0))_i$ is a basis of $\prod _i M_i$ over L. Indeed, because each $\{g_i(f_m^0)\}_m$ is a basis of $e_i \cdot M$ over $L_i$ , the set $\{f_m\}_m$ is linearly independent and (using crucially that $\{f_m\}_m$ is finite) the set $\{f_m\}_m$ also spans M.

Proof. For injectivity, as X is a H-stable disjoint union of $\{X_{{\mathfrak o}}\}_{{\mathfrak o}}$ , the morphism (3.1) factors as the product of morphisms

$$\begin{align*}\mathcal{O}_{X_{{\mathfrak o}}} \otimes_{L_{{\mathfrak o}}} \operatorname{\mathrm{Hom}}_{{\mathcal B}_{{\mathfrak o}}}(\mathcal{O}_{X_{{\mathfrak o}}}, {\mathcal V}_{{\mathfrak o}}) \rightarrow {\mathcal V}_{{\mathfrak o}}, \end{align*}$$

where , and we are reduced to showing that each is injective, for which we modify the proof of [Reference Taylor24, Lem. 4.3]. Recall that $L_{{\mathfrak o}}$ is a field by Lemma 3.2, and suppose that $f_1, ... ,f_k \in \operatorname {\mathrm {Hom}}_{{\mathcal B}_{{\mathfrak o}}}(\mathcal {O}_{X_{{\mathfrak o}}}, {\mathcal V}_{{\mathfrak o}})$ are $L_{{\mathfrak o}}$ -linearly independent. Define $e_1, ... , e_k \in {\mathcal V}_{{\mathfrak o}}(X_{{\mathfrak o}})^H$ by which are also $L_{{\mathfrak o}}$ -linearly independent because the natural map

$$\begin{align*}\operatorname{\mathrm{Hom}}_{{\mathcal B}_{{\mathfrak o}}}(\mathcal{O}_{X_{{\mathfrak o}}}, {\mathcal V}_{{\mathfrak o}}) \rightarrow {\mathcal V}(X_{{\mathfrak o}})^H, \qquad f \mapsto f(1_{X_{{\mathfrak o}}}) \end{align*}$$

is $L_{{\mathfrak o}}$ -linear and injective. Because $\mathcal {O}_{X_{{\mathfrak o}}}$ is irreducible in ${\mathcal B}_{{\mathfrak o}}$ by assumption (4), it is sufficient for us to show that the sum

$$\begin{align*}\sum_{i = 1}^k \mathcal{O}_{X_{{\mathfrak o}}} \cdot e_i \hookrightarrow {\mathcal V}_{{\mathfrak o}} \end{align*}$$

is direct. We prove this by induction on $k \geq 1$ . When $k = 1$ this is trivially true, so suppose that the statement is true for some fixed $k \geq 1$ , and consider

$$\begin{align*}\sum_{i = 1}^{k+1} \mathcal{O}_{X_{{\mathfrak o}}} \cdot e_i. \end{align*}$$

After rearranging the factors if necessary, it is sufficient to show that

$$\begin{align*}\left( \bigoplus_{i = 1}^{k} \mathcal{O}_{X_{{\mathfrak o}}} \cdot e_i \right) \bigcap \mathcal{O}_{X_{{\mathfrak o}}} \cdot e_{k+1} = 0. \end{align*}$$

If this intersection were non-zero, then

$$\begin{align*}\left( \bigoplus_{i = 1}^{k} \mathcal{O}_{X_{{\mathfrak o}}} \cdot e_i \right) \bigcap \mathcal{O}_{X_{{\mathfrak o}}} \cdot e_{k+1} = \mathcal{O}_{X_{{\mathfrak o}}} \cdot e_{k+1}, \end{align*}$$

by the irreducibility of $\mathcal {O}_{X_{{\mathfrak o}}} \cdot e_{k+1}$ . We can therefore write

$$\begin{align*}e_{k+1} = \lambda_1 e_1 + \cdots + \lambda_k e_k \end{align*}$$

for unique $\lambda _j \in \mathcal {O}(X_{{\mathfrak o}})$ . For any $h \in H$ ,

$$\begin{align*}0 = h(e_{k+1}) - e_{k+1} = (h(\lambda_1) - \lambda_1) e_1 + \cdots + (h(\lambda_k) - \lambda_k) e_k, \end{align*}$$

and therefore each $\lambda _j \in \mathcal {O}(X_{{\mathfrak o}})^H$ by induction. Furthermore, in case (A), for any affinoid (resp. affine) open subset U and $\partial \in {\mathcal T}(U)$ ,

$$ \begin{align*} 0 &= \partial(e_{k+1}) = \partial(\lambda_1 e_1 + \cdots + \lambda_k e_k), \\ &= \partial(\lambda_1)e_1 + \cdots + \partial(\lambda_k) e_k, \end{align*} $$

in ${\mathcal V}(U)$ , and therefore by induction $\partial (\lambda _i) = 0$ for all $i = 1, ... ,n$ . As this holds for each admissible open subset $U \subset X_{{\mathfrak o}}$ , each $\lambda _i \in c(X_{{\mathfrak o}})$ by [Reference Taylor24, Lem. 3.2]. Therefore, in either case (A) or (B), $\lambda _i \in L_{{\mathfrak o}}$ , which is a contradiction as we know that $e_1, ... ,e_k, e_{k+1}$ are linearly independent over $L_{{\mathfrak o}}$ .

Finally, the fact that $\operatorname {\mathrm {Hom}}_{{\mathcal B}}(\mathcal {O}_X, {\mathcal V})$ is free of finite rank over L follows directly from Lemma 2.2, which uses Lemma 3.2, that

$$\begin{align*}\operatorname{\mathrm{Hom}}_{{\mathcal B}}(\mathcal{O}_{X}, {\mathcal V}) \xrightarrow{\sim} \prod_{{\mathfrak o}} \operatorname{\mathrm{Hom}}_{{\mathcal B}_{{\mathfrak o}}}(\mathcal{O}_{X_{{\mathfrak o}}}, {\mathcal V}_{{\mathfrak o}}), \end{align*}$$

that each $\dim _{L_{{\mathfrak o}}} \operatorname {\mathrm {Hom}}_{{\mathcal B}_{{\mathfrak o}}}(\mathcal {O}_{X_{{\mathfrak o}}}, {\mathcal V}_{{\mathfrak o}}) \leq \text {rank}_{X_{{\mathfrak o}}}({\mathcal V}_{{\mathfrak o}}) < \infty $ , and that G acts transitively on the orbits ${\mathfrak o}$ .

Proof. For point (1), first note that $\mathcal {O}_X \otimes _L -$ is clearly exact, monoidal and faithful. To show that it is full, suppose that $V,W \in \operatorname {\mathrm {Rep}}_L^H(G)$ , and

$$\begin{align*}f \colon \mathcal{O}_X \otimes_L V \rightarrow \mathcal{O}_X \otimes_L W \end{align*}$$

is a morphism in ${\mathcal A}$ . For any $v \in V$ , because H acts trivially on V and W and f is H-equivariant,

$$\begin{align*}f_X(1_X \otimes v) \in \mathcal{O}(X)^H \otimes_L W. \end{align*}$$

Furthermore, in case (A), for any admissible open subset $U \subset X$ and $\partial \in {\mathcal T}(U)$

$$\begin{align*}\partial (f_U(1_U \otimes v)) = f_U(\partial(1_U) \otimes v) = 0, \end{align*}$$

and therefore

$$\begin{align*}f_X(1_X \otimes v) \in c(X) \otimes_L W \end{align*}$$

by [Reference Taylor24, Lem. 3.2]. In either case (A) or (B),

$$\begin{align*}f_X(1_X \otimes v) \in L \otimes_L W \equiv W, \end{align*}$$

and this allows us to define a morphism $\lambda \colon V \rightarrow W$ with $f_X(1_X \otimes v) = 1 \otimes \lambda (v)$ . It is direct to see, as f is a morphism in ${\mathcal A}$ , that $f = 1 \otimes \lambda $ and hence $\mathcal {O}_X \otimes _L -$ is full.

For point (2), first note that because H is normal in G we may view $\operatorname {\mathrm {Hom}}_{{\mathcal B}}(\mathcal {O}_X, -)$ as a functor from ${\mathcal A}$ to $\operatorname {\mathrm {Rep}}_L^H(G)$ , not just ${\mathbf {Mod}}_L$ , as described above. If ${\mathcal V} \in {\mathcal A}$ and the injective map (3.1) is an isomorphism then ${\mathcal V}$ is in the essential image of the functor $\mathcal {O}_X \otimes _L -$ , and on the full subcategory of such ${\mathcal V}$ the functor $\operatorname {\mathrm {Hom}}_{{\mathcal B}}(\mathcal {O}_X, -)$ provides a right quasi-inverse to $\mathcal {O}_X \otimes _L -$ . Conversely, if ${\mathcal V} = \mathcal {O}_X \otimes _L V$ is in the essential image, then

$$\begin{align*}\operatorname{\mathrm{Hom}}_{{\mathcal B}}(\mathcal{O}_X, \mathcal{O}_X \otimes_L V) = \operatorname{\mathrm{Hom}}_{{\mathcal B}}(\mathcal{O}_X \otimes_L L, \mathcal{O}_X \otimes_L V) \xleftarrow{\sim} \operatorname{\mathrm{Hom}}_{L[H]}(L, V) \equiv V, \end{align*}$$

by the fully faithfulness of point (1) in the case when we take $G = H$ . Therefore,

$$\begin{align*}\mathcal{O}_X \otimes_L \operatorname{\mathrm{Hom}}_{{\mathcal B}}(\mathcal{O}_X, {\mathcal V}) \rightarrow {\mathcal V} \end{align*}$$

is an isomorphism, being identified with the identity map

$$\begin{align*}\mathcal{O}_X \otimes_L V \rightarrow {\mathcal V} \end{align*}$$

and we see that $\operatorname {\mathrm {Hom}}_{{\mathcal B}}(\mathcal {O}_X, -)$ provides a left quasi-inverse to $\mathcal {O}_X \otimes _L -$ .

For point (3), in case (A) one can use the same argument as [Reference Taylor24, Thm. 4.4(4)], which uses point (2) and the fact that ${\mathbf {VectCon}}^G(X)$ is closed under quotients in ${\mathbf {Mod}}(G\text {-}{\mathcal D}_X)$ . However, it is not true in general that ${\mathbf {Vect}}^G(X)$ is closed under quotients in ${\mathbf {Mod}}(G \text {-}\mathcal {O}_X)$ , and so we give an alternative proof. First note that because the functor is exact and full, it is sufficient to show that the essential image is closed under sub-objects. Given $V \in \operatorname {\mathrm {Rep}}^H_L(G)$ , suppose that ${\mathcal F} \subset \mathcal {O}_X \otimes _L V$ is a sub-object in ${\mathcal A}$ , and define

This is an L-submodule of V as the tensor product is over L, and further a $L \rtimes G$ - submodule of V, as given $g \in G$ and $v \in V$ , then for any open subset $U \subset X$ and $s \in \mathcal {O}_X(U)$ ,

$$\begin{align*}s \otimes g(v) = g^{\mathcal{O}_X}_{g^{-1}(U)}((g^{-1})^{\mathcal{O}_X}_U(s)) \otimes g(v) = g^{\mathcal{O}_X \otimes V}((g^{-1})^{\mathcal{O}_X}_U(s) \otimes v) \in {\mathcal F}(U). \end{align*}$$

We now claim that the injection $\mathcal {O}_X \otimes _L W \hookrightarrow {\mathcal F}$ is an isomorphism. It is sufficient to show that for all $i \in \pi _0(X)$ , the restriction to $X_i$ is surjective, or equivalently that the morphism on sections

$$\begin{align*}\mathcal{O}(X_i) \otimes_{L} W \hookrightarrow {\mathcal F}(X_i) \end{align*}$$

is surjective, as each $X_i$ is quasi-Stein. Furthermore, as $e_{{\mathfrak o}}$ acts as the identity on $\mathcal {O}(X_i)$ , the inclusion $e_{{\mathfrak o}} \cdot W \hookrightarrow W$ induces an isomorphism

$$\begin{align*}\mathcal{O}(X_i) \otimes_{L_{{\mathfrak o}}} e_{{\mathfrak o}} \cdot W \xrightarrow{\sim} \mathcal{O}(X_i) \otimes_{L} W. \end{align*}$$

The same is true for V, and we have a commutative diagram

where we have defined $M_i$ to be the preimage of ${\mathcal F}(X_i)$ in $\mathcal {O}(X_i) \otimes _{L_{{\mathfrak o}}} e_{{\mathfrak o}} \cdot V$ . Suppose that

$$\begin{align*}z = \sum_{k} s_k \otimes v_k \in M_i \subset \mathcal{O}(X_i) \otimes_{L_{{\mathfrak o}}} e_{{\mathfrak o}} \cdot V, \end{align*}$$

noting that without loss of generality we may assume that the elements $s_k$ are $L_{i}$ -linearly independent by choosing a $L_{i}$ basis of the $L_{i}$ -span of $\{s_k\}_k$ and using the isomorphism $L_{{\mathfrak o}} \xrightarrow {\sim } L_i$ .

By assumption (4’) $\mathcal {O}_{X_i}$ is irreducible in ${\mathcal B}_i$ , and thus $\mathcal {O}(X_i)$ is irreducible as an R-module (where $R = {\mathcal D}(X_i) \rtimes H_i$ in case (A) and $R = \mathcal {O}(X_i) \rtimes H_i$ in case (B)) by [Reference Taylor24, Prop. 2.44] as $X_i$ is quasi-Stein. Furthermore

$$\begin{align*}\operatorname{\mathrm{End}}_{R}(\mathcal{O}(X_i)) = L_i \end{align*}$$

by the definition of $L_i$ . Therefore, for any index j we may apply the Jacobson Density Theorem to assert the existence of some $x_j \in R$ such that $x_j$ acts on the $L_i$ -span of $\{ s_k \}_k$ by the projection to the basis element $s_j$ . Because $H_i$ acts trivially on $e_{{\mathfrak o}} \cdot V$ , R only acts on the first factor of $\mathcal {O}(X_i) \otimes _{L_{{\mathfrak o}}} e_{{\mathfrak o}} \cdot V$ , and therefore $x_j(z) = s_j \otimes v_j \in {\mathcal F}(X_i)$ , and thus $1 \otimes v_j \in {\mathcal F}(X_i)$ , using again that $\mathcal {O}(X_i)$ is a simple R-module. But then also $\mathcal {O}_{X_{{\mathfrak o}}} \otimes v_j \subset {\mathcal F}|_{X_{{\mathfrak o}}}$ as $\mathcal {O}_{X_{{\mathfrak o}}}$ is irreducible in ${\mathcal B}_{{\mathfrak o}}$ and H acts trivially on V. Therefore, each $v_j \in e_{{\mathfrak o}} \cdot W$ , as we can describe $e_{{\mathfrak o}} \cdot W$ as

$$\begin{align*}e_{{\mathfrak o}} \cdot W = \{v \in V \mid e_{{\mathfrak o}} \cdot v = v \text{ and } s \otimes v \in {\mathcal F}(U) \text{ for any open } U \subset X_{{\mathfrak o}}, s \in \mathcal{O}_{X_{{\mathfrak o}}}(U)\}. \end{align*}$$

In particular, $z \in \mathcal {O}(X_i) \otimes _{L_{{\mathfrak o}}} e_{{\mathfrak o}} \cdot W$ and $\mathcal {O}(X_i) \otimes _{L} W \hookrightarrow {\mathcal F}(X_i)$ is surjective.

Finally, to see that W is free of finite rank over L, as G acts transitively on the orbits ${\mathfrak o}$ , and $\dim _{L_{{\mathfrak o}}} e_{{\mathfrak o}} \cdot W \leq \dim _{L_{{\mathfrak o}}} e_{{\mathfrak o}} \cdot V < \infty $ , we may deduce this from Lemma 2.2 once we know that the map

$$\begin{align*}W \rightarrow \prod_{{\mathfrak o}} e_{{\mathfrak o}} \cdot W \end{align*}$$

is an isomorphism. This is injective, as the corresponding map for V is an isomorphism, V being free over L. For surjectivity, given $(w_{{\mathfrak o}})_{{\mathfrak o}}$ , from the isomorphism for V there is a unique $v \in V$ with $e_{{\mathfrak o}} \cdot v = w_{{\mathfrak o}}$ for all ${\mathfrak o} \in I$ . To see that $v \in W$ , suppose that $U \subset X$ is an admissible open subset, and $s = (s_{{\mathfrak o}})_{{\mathfrak o}} \in \prod _{{\mathfrak o}} \mathcal {O}_{X_{\mathfrak o}}(U \cap X_{{\mathfrak o}}) = \mathcal {O}_X(U)$ . Then

$$ \begin{align*} s \otimes v &= (s_{{\mathfrak o}} \otimes v)_{{\mathfrak o}}, \\ &= (e_{{\mathfrak o}} \cdot s_{{\mathfrak o}} \otimes v)_{{\mathfrak o}}, \\ &= (s_{{\mathfrak o}} \otimes e_{{\mathfrak o}} \cdot v)_{{\mathfrak o}}, \\ &= (s_{{\mathfrak o}} \otimes m_{{\mathfrak o}})_{{\mathfrak o}} \end{align*} $$

which lies in $\prod _{{\mathfrak o}} {\mathcal F}(U \cap X_{{\mathfrak o}}) = {\mathcal F}(U)$ and therefore $v \in W$ .

Proof. All that remains to show is that if $\text {rank}_L V^H = \text {rank}_F V$ , then (3.2) is surjective. Writing W for the image of (3.2), it is sufficient to show that the quotient $V / W$ is free, as then $V \cong W \oplus V / W$ and hence $V / W$ must have rank $0$ . For this, by Lemma 2.2 it is sufficient to show that the natural map $V/W \xrightarrow {\sim } \prod _i e_i \cdot (V/W)$ is an isomorphism, which follows directly from the isomorphisms $V \xrightarrow {\sim } \prod _i e_i \cdot V$ and $W \xrightarrow {\sim } \prod _i e_i \cdot W$ .

Proof. The action of $G^0$ on ${\mathcal N}_m$ is continuous in the sense of [Reference Ardakov1, Def. 3.1.8] by [Reference Taylor24, Cor. 7.1], and therefore the stabiliser of U in $G^0$ is open and thus contains some $G_{s}$ . Set , and let $\rho \colon G_s \rightarrow \operatorname {\mathrm {Aut}}_K(A)$ denote the action map. Choosing a formal model ${\mathcal A}$ of A, this defines a topology on $\operatorname {\mathrm {Aut}}_K(A)$ [Reference Ardakov1, Def. 3.1.3] which is independent of the choice of formal model ${\mathcal A}$ of A [Reference Ardakov1, Thm. 3.1.5]. The open subgroup ${\mathcal G}_{p^{2}}({\mathcal A}) \subset \operatorname {\mathrm {Aut}}_K(A)$ has the property that for any $\varphi \in {\mathcal G}_{p^{2}}({\mathcal A})$ the series $\log (\varphi )$ converges and defines an element of $\operatorname {\mathrm {Der}}_{{\mathcal R}}({\mathcal A})$ , where ${\mathcal R}$ is the ring of integers of K.

The continuity of the action of $G^0$ on ${\mathcal N}_m$ further implies that $\rho \colon G_s \rightarrow \operatorname {\mathrm {Aut}}_K(A)$ is continuous, and thus there is some $G_r \leq G_s$ with $\rho (G_r) \subset {\mathcal G}_{p^{2}}({\mathcal A})$ . We therefore have that $\log (g^{\mathcal {O}}_U)$ converges for any $g \in G_r$ to an element of $\operatorname {\mathrm {Der}}_{\mathcal R}({\mathcal A})$ , which we consider as a derivation of A via the canonical inclusion $\operatorname {\mathrm {Der}}_{\mathcal R}({\mathcal A}) \hookrightarrow \operatorname {\mathrm {Der}}_K(A)$ (noting that $A = {\mathcal A}[1/p]$ ).

For the third claim, fix an affine chart ${\mathbb A}_K^{n-1, \text {an}} \subset {\mathbb P}_K^{n-1, \text {an}}$ with $\Omega \subset {\mathbb A}_K^{n-1, \text {an}}$ , and let $\{\Omega _j\}_{j \geq 1}$ be the quasi-Stein open covering of $\Omega $ defined by setting $\Omega _j = \Omega \cap {\mathbb D}_j$ , the intersection in ${\mathbb A}_K^{n-1, \text {an}}$ of $\Omega $ with the affinoid $(n-1)$ -dimensional ball ${\mathbb D}_j = \{x \in {\mathbb A}_K^n \mid |x| \leq j\}$ of centre $0 \in {\mathbb A}_K^{n-1, \text {an}}$ and radius j. Let $\{U_j\}_{j \geq 1}$ be the induced quasi-Stein open covering of ${\mathcal N}_m$ , where $U_j$ is the preimage of $\Omega _j$ in ${\mathcal N}_m$ under the $G_0$ -equivariant finite étale map ${\mathcal N}_m \rightarrow {\mathcal N} \xrightarrow {\sim } \Omega $ .

Because U is quasi-compact, U is contained in some $U_j$ for some $j \geq 1$ , and the composition

$$\begin{align*}U \hookrightarrow U_j \rightarrow \Omega_j \hookrightarrow {\mathbb D}_j \end{align*}$$

is étale and $G^0$ -equivariant. The tangent sheaf ${\mathcal T}({\mathbb D}_j)$ is generated over $\mathcal {O}({\mathbb D}_j)$ by the derivations

for coordinates $x_1, ... ,x_{n-1}$ of ${\mathbb A}^{n-1, \text {an}}$ . By the same argument as above, there is some $G_k$ such that $G_k$ stabilises ${\mathbb D}_j$ and $\log (g^{\mathcal {O}}_{{\mathbb D}_j})$ converges for any $g \in G_k$ which (after potentially going to a smaller congruence subgroup) we may assume is the same $G_k$ as above. From the formula for the logarithm,

$$\begin{align*}\pi^k \partial_i = \log(g^{\mathcal{O}}_{i,k, {\mathbb D}_j}) \end{align*}$$

where $g_{i,k} \in G_k$ is the elementary matrix in $G_k$ which acts by the Möbius transformation which sends $x_i \mapsto x_i + \pi ^k$ , and fixes each other $x_j$ .

By [Reference Taylor24, Lem. 2.12], because the morphism $U \rightarrow {\mathbb D}_j$ is étale, for each i there is a unique derivation $\partial ^{\prime }_i$ of $\mathcal {O}(U)$ which restricts to $\partial _i$ on the image of $\mathcal {O}({\mathbb D}_j)$ in $\mathcal {O}(U)$ , and these generate ${\mathcal T}(U)$ as an $\mathcal {O}(U)$ -module because the derivations $\partial _i$ generate ${\mathcal T}({\mathbb D}_j)$ as an $\mathcal {O}({\mathbb D}_j)$ -module. Because the morphism $\mathcal {O}({\mathbb D}_j) \rightarrow \mathcal {O}(U)$ is continuous (being a morphism of affinoid algebras), $\log (g^{\mathcal {O}}_{i,k,U})$ restricts to $\log (g^{\mathcal {O}}_{i,k,{\mathbb D}_j})$ on the image of $\mathcal {O}({\mathbb D}_j)$ in $\mathcal {O}(U)$ . In particular, from the uniqueness of $\partial ^{\prime }_i$ described above, $\partial ^{\prime }_i$ is explicitly given by

$$\begin{align*}\partial^{\prime}_i = \pi^{-k} \log(g^{\mathcal{O}}_{i,k,U}), \end{align*}$$

and therefore $\log (G_k)$ generates ${\mathcal T}(U)$ as an $\mathcal {O}(U)$ -module.

Proof. To prove statement (1), suppose that ${\mathcal V}$ is a non-trivial $G_m \text {-}\mathcal {O}_{X}$ -submodule of $\mathcal {O}_{X}$ . We show that ${\mathcal V}$ is in fact a ${\mathcal D}_{X}$ -submodule of $\mathcal {O}_{X}$ , from which it follows that ${\mathcal V} = \mathcal {O}_{X}$ as $\mathcal {O}_{X}$ is irreducible as a ${\mathcal D}_{X}$ -module because X is connected [Reference Taylor24, Lem. 2.37].

To this end, let $U \subset X$ be a connected admissible open affinoid subset. By Lemma 4.1, we can find some $G_k \subset G_m$ such that $G_k$ stabilises U and $\log (G_k)$ generates ${\mathcal T}(U)$ as an $\mathcal {O}(U)$ -module, and thus it is sufficient to show that for any $g \in G_k$ and $v \in {\mathcal V}(U)$ , $\log (g_U^{\mathcal {O}})(v) \in {\mathcal V}(U)$ . Because $\mathcal {O}(U)$ is a noetherian Banach algebra, ${\mathcal V}(U)$ is closed in the subspace topology [Reference Schneider and Teitelbaum22, Prop. 2.1], and therefore

$$\begin{align*}\log(g_U^{\mathcal{O}})(v) = \sum_{k = 1}^{\infty} \frac{(-1)^{k+1}}{k} (g_U^{\mathcal{O}} - 1)^k(v) \in {\mathcal V}(U), \end{align*}$$

as each $(g_U^{\mathcal {O}} - 1)^k(v) \in {\mathcal V}(U)$ by the assumption that ${\mathcal V} \subset \mathcal {O}_X$ is a $G_m \text {-}\mathcal {O}_{X}$ -submodule.

Proof. For $m \geq 1$ , suppose that ${\mathcal V} \in {\mathbf {Vect}}^{G_0}_{\operatorname {\mathrm {LT}}}(\Omega )$ . As all spaces are quasi-Stein, it is sufficient to give the isomorphism on global sections. Writing $M = \Gamma (\Omega , {\mathcal V})$ , then in the notation of [Reference Kohlhaase13],

$$ \begin{align*} \Gamma(\Omega, {\mathbb D}_{\operatorname{\mathrm{LT}}}({\mathcal V})) &= (C_m \otimes_{A_0} M)^{G_0}, \\ &\equiv (B_m \hat{\otimes}_{K_m} A_m \otimes_{A_0} M)^{G_0},\\ &\equiv (B_m \hat{\otimes}_{K_m} (A_m \otimes_{A_0} M)^{G_m})^{G_0 / G_m}, \\ &\equiv (B_m \hat{\otimes}_{K_m} (A_m \otimes_{K_m} V)^{G_m})^{G_0 / G_m}. \end{align*} $$

Here for the third equality we use [Reference Kohlhaase13, Lem. 3.3] and the fact that $G_m$ acts trivially on $B_m$ . In particular, if ${\mathcal V} = \Phi ^0_{\operatorname {\mathrm {LT}},m}(V)$ , then $A_m \otimes _{A_0} M = A_m \otimes _{K_m} V$ and

$$ \begin{align*} B_m \hat{\otimes}_{K_m} (A_m \otimes_{A_0} M)^{G_m} &= B_m \hat{\otimes}_{K_m} (A_m \otimes_{K_m} V)^{G_m}, \\ &= B_m \hat{\otimes}_{K_m} (K_m \otimes_{K_m} V), \\ &\equiv B_m \otimes_{K_m} V, \end{align*} $$

using that $G_m$ acts trivially on V, and thus there is a natural isomorphism

$$ \begin{align*} \Gamma(\Omega, {\mathbb D}_{\operatorname{\mathrm{LT}}}(\Phi^0_{\operatorname{\mathrm{LT}},m}(V))) &\equiv (B_m \otimes_{K_m} V)^{G_0 / G_m}, \\ &= \Gamma(\Omega, \Phi^0_{\operatorname{\mathrm{Dr}},m}(V)). \\[-38pt] \end{align*} $$

Proof. Each category $\operatorname {\mathrm {Rep}}_{K_m}^{Z_m}(G_0 \times H_0)$ is semisimple, by [Reference Montgomery17, Cor. 0.2]. Each inclusion

$$\begin{align*}K_{\infty} \otimes_{K_m} - \colon \operatorname{\mathrm{Rep}}_{K_m}^{Z_m}(G_0 \times H_0) \xrightarrow{\sim} \operatorname{\mathrm{Rep}}_{K_{\infty}}^m(G_0 \times H_0) \hookrightarrow \operatorname{\mathrm{Rep}}^{\operatorname{\mathrm{sm}}}_{K_{\infty}}(G_0 \times H_0) \end{align*}$$

is closed under sub-quotients by Lemma 5.3 and thus $\operatorname {\mathrm {Rep}}^{\operatorname {\mathrm {sm}}}_{K_{\infty }}(G_0 \times H_0)$ is semisimple.

Proof. From the explicit description (5.3) above, if $G_m$ acts trivially on V, then

$$ \begin{align*} \Psi_{G}^0(V) &= (\mathcal{O}_{{\mathcal N}_m} \otimes_K V)^{H_0/H_m}, \\ &= \mathcal{O}_{{\mathcal N}_m}^{H_0/H_m} \otimes_K V, \\ &= \mathcal{O}_{\Omega} \otimes_K V \end{align*} $$

because $H_0$ acts trivially on V. The claim for $\Psi _{H}^0(W)$ follows similarly using Proposition 5.5.

Proof. This follows directly from Corollary 3.9 with triples $(K_{\infty }, G_0 \times H_0,H_0)$ and $(K_{\infty }, G_0 \times H_0,G_0)$ respectively, as $K_{\infty }^{G_0} = K = K_{\infty }^{H_0}$ .

Proof. The functor $K_{\infty } \otimes _K -$ is the direct limit of the functors

$$\begin{align*}K_m \otimes_K - \colon \operatorname{\mathrm{Rep}}_K(G_0 / G_m) \rightarrow \operatorname{\mathrm{Rep}}_{K_m}(G_0 / G_m \times H_0/H_m), \end{align*}$$

which are equivalences with inverse $(-)^{H_0 / H_m}$ when $n = 1$ by Galois descent [Reference Taylor24, Prop. 2.53], as $H_0 / H_m$ is the Galois group of $K_m / K$ . The same holds similarly for $G_0$ and $H_0$ swapped.

Proof. Fix an object $V \in \operatorname {\mathrm {Rep}}_K^{\operatorname {\mathrm {sm}}}(G_0)$ , and let $m \geq 1$ be such that $V^{G_m} = V$ . We take L to be a finite extension of K which is a splitting field for the group , and set (which has an action of $G_0 \times H_0$ which is trivial on the first component). We first note that by Theorem 5.4 and Lemma 5.3, the following statements are equivalent:

• • $\Psi _G^0(V)$ is in the essential image of $\Psi _H^0 \colon \operatorname {\mathrm {Rep}}_K^{\operatorname {\mathrm {sm}}}(H_0) \rightarrow {\mathbf {Vect}}^{G_0}_{\operatorname {\mathrm {LT}}}(\Omega )$ ,

• • $K_{\infty } \otimes _K V$ is in the essential image of $K_{\infty } \otimes _K - \colon \operatorname {\mathrm {Rep}}_K^{\operatorname {\mathrm {sm}}}(H_0) \rightarrow \operatorname {\mathrm {Rep}}_{K_{\infty }}^{\operatorname {\mathrm {sm}}}(G_0 \times H_0)$ ,

• • $K_m \otimes _K V$ is in the essential image of $K_{m} \otimes _K - \colon \operatorname {\mathrm {Rep}}_K^{H_m}(H_0) \rightarrow \operatorname {\mathrm {Rep}}_{K_m}^{Z_m}(G_0 \times H_0)$ .

We are therefore reduced to showing that this final condition is equivalent to V being inflated from $\operatorname {\mathrm {Rep}}_K^{\operatorname {\mathrm {sm}}}(\mathcal {O}_F^\times )$ . We first consider $V_L$ . Taking the triple $(F,G,H)$ in Corollary 3.8 and Corollary 3.9 to be $(L_m, G_0 \times H_0, G_0)$ , this essential image of the functor

$$\begin{align*}L_m \otimes_L - \colon \operatorname{\mathrm{Rep}}_L^{H_m}(H_0) \rightarrow \operatorname{\mathrm{Rep}}^{Z_m}_{L_{m}}(G_0 \times H_0), \end{align*}$$

consists of exactly those U for which the inequality

$$\begin{align*}\dim_L U^{G_0/G_m} \leq \dim_{L_m}(U) \end{align*}$$

is an equality. Taking $U = L_m \otimes _L V_L$ this inequality becomes

(5.4) $$ \begin{align} \dim_L(L_m \otimes_L V_L)^{G_0/ G_m} \leq \dim_K V_L. \end{align} $$

By the normal basis theorem [Reference Chase, Harrison and Rosenberg8, Thm. 4.2(c)] we have that $L_m \cong L[A]$ as $L[A]$ -modules because $L_m/L$ is Galois with Galois group A. Writing $V_L$ as the direct sum of irreducible representations $V_i$ , the inequality (5.4) is an equality if and only it is for each $V_i$ , and, by our assumption that L is a splitting field of A, (5.4) is an equality for a fixed $V_i$ if and only if some twist of $V_i$ by an inflated character of A is trivial, or equivalently that $V_i$ is an inflated character of A.

In summary, $L_m \otimes _L V_L$ is in the essential image of the functor $L_m \otimes _L -$ above if and only if $V_L$ is inflated from a smooth representation of $\mathcal {O}_F^\times $ . In particular, if $K_m \otimes _K V \cong K_m \otimes _K W$ for some $W \in \operatorname {\mathrm {Rep}}_K^{H_m}(H_0)$ , then $V_L \cong W_L$ and $V_L$ is in the essential image of $L_m \otimes _L -$ , so $V_L$ and hence V is inflated from a smooth representation of $\mathcal {O}_F^\times $ . Conversely, suppose that V is inflated from a smooth representation of $\mathcal {O}_F^\times $ . Then $V_L$ is too and hence $L_m \otimes _L V_L$ is in the essential image of $L_m \otimes _L -$ . We claim that $K_m \otimes V \cong K_m \otimes W$ for . Indeed, the natural map

$$\begin{align*}K_m \otimes_K W \rightarrow V \end{align*}$$

is an isomorphism, because after applying $L \otimes _K -$ it becomes the map

$$\begin{align*}L_m \otimes_L (L_m \otimes_L V_L)^{G_0 / G_m} \rightarrow L_m \otimes_L V_L, \end{align*}$$

which is an isomorphism by Corollary 3.8 and Corollary 3.9.